The 17-year-old Anfield winger has emerged as one of England's brightest prospects since breaking into the Liverpool first-team, and recent reports have linked United with a move for the England international.
Morgan, an ex-Scotland international who made 236 appearances for United in the 1960s and 1970s, thinks Sterling has the ability to become one of the best wingers in the world.
Speaking to the Manchester Evening News, Morgan said: "I think United should attempt to buy Liverpool's left winger Raheem Sterling.
"Sterling is the first winger I have seen in a long time who looks like a proper winger. He looks very good already and I think he is going to be great.
"If you just let him go wide on the outside of the fullbacks all the time I think he will be a great player. He'll frighten people to death, so long as nobody coaches it out of him. He's the brightest prospect out there and would suit United."
No player has moved straight from Liverpool to United, or United to Liverpool, since 1964 - Phil Chisnall was the last to cross the divide.
Liverpool boss Brendan Rodgers has given Sterling an extended run in the team, and the teenage winger has rewarded the Anfield boss with a string of fine performances.
His displays on the Liverpool wing caught the eye of England manager Roy Hodgson, and Sterling won his first full England cap during last month's friendly in Sweden.
But on-going negotiations over his contract have led to speculation Manchester United could stage a sensational swoop for the Liverpool star in January.
Rodgers is thought to be desperate to keep hold of Sterling, and the report claims the Reds manager reminded him of the chances afforded to Sterling at Anfield.
But talks between the youngster's advisors and Liverpool managing director Ian Ayre are believed to have stalled, potentially opening the door for United.